#368104

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14015
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1531 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12,8,14,6,13)(3,9,17,32,18,33,28,31)(4,19,23,10,22,37,24,36)(5,11,25,39,27,40,26,38)(15,34,20,48,42,57,43,29)(16,30,45,55,46,56,44,58)(21,50,53,60,51,47,52,35)(41,61,49,54,62,59,63,64);(1,3,7,17,8,18,6,28)(2,9,12,32,14,33,13,31)(4,20,23,42,22,43,24,15)(5,16,25,45,27,46,26,44)(10,34,37,48,36,57,19,29)(11,30,39,55,40,56,38,58)(21,49,53,62,51,63,52,41)(35,59,50,64,60,61,47,54);(1,4,6,24,8,22,7,23)(2,10,13,19,14,36,12,37)(3,15,17,20,18,42,28,43)(5,21,26,52,27,51,25,53)(9,29,32,34,33,48,31,57)(11,35,38,47,40,60,39,50)(16,41,45,49,46,62,44,63)(30,54,55,59,56,64,58,61);(1,5)(2,11)(3,16)(4,21)(6,26)(7,25)(8,27)(9,30)(10,35)(12,39)(13,38)(14,40)(15,41)(17,45)(18,46)(19,47)(20,49)(22,51)(23,53)(24,52)(28,44)(29,54)(31,58)(32,55)(33,56)(34,59)(36,60)(37,50)(42,62)(43,63)(48,64)(57,61)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317339, #72317395
maximal normal subgroup of	#72317339, #72317395

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275300);